Question

what are the solutions to the equation 3(x - 5)² = 27?
a ( x = -2, x = 8 )
b ( x = 4, x = 14 )
c ( x = 2, x = 8 )
d ( x = -5, x = 5 )

Explanation:

Step1: Divide both sides by 3

To isolate the squared term, divide each side of the equation \( 3(x - 5)^2 = 27 \) by 3.
\( \frac{3(x - 5)^2}{3} = \frac{27}{3} \)
Simplifying gives \( (x - 5)^2 = 9 \).

Step2: Take square root of both sides

Take the square root of both sides. Remember that taking the square root of a number gives two solutions: positive and negative.
\( \sqrt{(x - 5)^2} = \pm\sqrt{9} \)
This simplifies to \( x - 5 = \pm 3 \).

Step3: Solve for x (two cases)

Case 1: Positive square root

If \( x - 5 = 3 \), add 5 to both sides:
\( x = 3 + 5 \)
\( x = 8 \).

Case 2: Negative square root

If \( x - 5 = -3 \), add 5 to both sides:
\( x = -3 + 5 \)
\( x = 2 \).

Answer:

\( x = 2, x = 8 \) (Option C, assuming C is labeled \( x = 2, x = 8 \))